Two things which (I think) are closely related and intertwined with the OP:

1. The body as an object.
It’s fairly well-established that human babies first see their body parts as things that are ‘part of the environment’, so to speak. The realization that the chubby hands and feet are somehow connected to them and controllable takes time.

So perhaps at least the body is viewed from the beginning by the person ‘as an object’, and somewhat remains at least partially an object, in addition to being part of a subject called ‘me’.

2. We are thinking of self as subject, object, both, or some other category (though I’m not sure what that’d be called). Identity is a concept integral with the self viewed from any perspective, I would think.

How does the identity / identities that arise, come and go relate to subjective self and objective self?

The idea of self being related to embodiment is especially true and that is where the mind/body problem seems to err with some viewing mind as a 'ghost in the machine. This kind of perspective may have fading with the knowledge of neuroscience and quantum theory making the idea of mind or self as an entity appear as rather ridiculous.

The idea of the updating of self is interesting because it may not that this is not done on a conscious basis mostly. The whole dimension of social experiences, as well as the body itself may affect people so much, with there being a narcissistic element to social life. It may be difficult not to be affected by others' perceptions of oneself at all. The core basis of self may begin in early childhood, as the object-relations theorists, such as Winnicott suggest, but it goes on and on. The maintenance of a balanced sense of self, as opposed to a fragmented one in many psychiatric disorders and psychological issues is central to wellbeing.

It does seem that most theories are not able to capture the nature of awareness itself, which is so complex an aspect of experience itself. The idea of 'witness' consciousness was suggested by Ken Wilber, and this can be seen as a third factor within the mind and body relationship. This nature of awareness, which goes back to Descartes' 'I', is dependent on reflection based synthesis, with language giving rise to narrative identity, including the significance of memories and ideas of potential future self as seen in the moment.

The experience of the body as object is important too. It probably takes some time before a person establishes being able to see one's body in this way. The idea of the sense of separation from the mother is meant to be an important marker. The experience of being able to see onesealf in the mirror may be a crucial point too because it enables looking at oneself in a detached way, like being able to view oneself from the outside rather than simply in the internal world of embodied experience.

The more I tried to think from the standpoint of idealized rational subjectivity the more I encounter the fact of social-embedment. It's all well and good to view the brain as some kind of idealized embodiment of consciousness, susceptible of scientific analysis. But the species as an organism is a much more compelling kind of entity to me. If there can be such as thing as a collective mind (which I think is self-evident, trivially and non-trivially) then it serves as a nice transition from personal to transcendental consciousness. The project becomes one of mutual self-objectification.

The "standpoint of idealized rational subjectivity", as described by Descartes, comes into being out of nothing. In the Meditations, he argues that the experience of isolated awareness and the choices available to it are not inherited from parents but are created moment to moment by God. That perspective does not favor any attempt to understand how the 'thinking activity' came into being as a process of nature. Descartes recognizes the embodiment is experienced directly but deliberately avoids treating the union as a development. That is quite a different idea from seeing "the brain as some kind of idealized embodiment of consciousness, susceptible of scientific analysis." When differentiating between the biological organism from the social environment the organism emerges from, it is not a matter of either/or selection between the factors that explains the 'self' experience.

Rejecting a 'brain in a vat' position is to look for a model where the possibility for social forms is made possible through evolution. How to distinguish between what is hard wired instinctively from what is capable of change and adaptation involves a larger view of ecology along with sharper methods of reduction. One aspect of that double movement that speaks to Mead's call for a multi-disciplinarian approach is the role of Developmental Immaturity. Some genetic processes are sped up in changes of species and others are slowed down. The importance of the concept of neoteny is important to both 'biological' and 'sociological' registers:

Contemporary evolutionary theorists no longer see evolution as progressive in the sense of developing toward ever-increasing levels of complexity (see Gould, 1989); nor is the biogenetic law taken seriously. Many aspects of evolution can be seen as additions or acceleradotls of a developmental trend but certainly not all and perhaps not even most. In many cases, important evolutionary changes are brought about by retardation of development, not by acceleration. This is reflected by the concept of neoteny, which means literally "holding youth" or the retention of embryonic or juvenile characteristics by a retardation of development. Neoteny is an example of the process of heterochrony—genetic-based differences in developmental timing, de Beer (1958) proposed that changes in the timing of ontogeny are the driving force of evolution, and many evolutionary biologists over the course of this century have concurred. — The Role of Immaturity in Human Development, by David F. Bjorklund

Whether through this formulation or another, immaturity permits a response to the environment rather than being hard wired to an 'innate' condition. Development psychology approaches that from many angles, from Jung wondering if we manipulate our instincts to Vygotsky understanding 'isolated rational subjectivity' as a skill learned through years of social interaction and training. For development of particular persons, immaturity is interrelated to the profound dependence upon care givers, the anxiety and fear as reflected in the Mirror Stage as depicted by Lacan. Such a model does not explain the Cartesian Theater but looks for events which places it somewhere.

:up:

In the Meditations, he argues that the experience of isolated awareness and the choices available to it are not inherited from parents but are created moment to moment by God. That perspective does not favor any attempt to understand how the 'thinking activity' came into being as a process of nature. — Paine

I don't know about you, but when I read something historical, I always make allowances for the limits of scientific understanding at the time. Yes, religion still overshadowed much reasoning at the time of the Enlightenment. In fact, a rebalancing of religion and humanism is a significant feature of the Enlightenment. So anytime a philosopher attributes a function to God, I usually restrict my intake to the functional aspects. Avatar aside, I did not have Descartes in mind when I made my comments.

The significance of the theological was to differentiate between the cogito as a given rather than an outcome of a natural process. I was wondering how taking that as a given relates to your saying: "view the brain as some kind of idealized embodiment of consciousness, susceptible of scientific analysis."

It seems to me that "idealized rational subjectivity," is abandoned by biology as well as by social sciences. The evolutionary view binds all the disciplines into the exploration of the same nature.